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IOSR Journal of Applied Physics (IOSR-JAP)
e-ISSN: 2278-4861.Volume 6, Issue 3 Ver. II (May-Jun. 2014), PP 38-41
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The Dependence of Spatial Distribution of Excitement of
Receptors on Their Stiles-Crawford Effect
Nachieketa K Sharma
Department of Physics, Institute of Technical Education & Research, Siksha ‘O’ Anusandhan University,
Khandagiri Square, Bhubaneswar, OR, India
Abstract: The spatial distribution of excitement of receptors depends on the magnitude of amplitude of light
integrated over the wavefront passing through the pupil as conditioned by the receptor’s acceptance lobe.
Because the peripheral zones of the pupil are less efficient in eliciting visual excitation than the pupil’s centre,
otherwise known as Stiles-Crawford effect of the first kind (SCE I), the spatial distribution of excitement of
receptors will differ depending on their Stiles- Crawford effect. Assuming the funneling of light into the
receptors through a single entering aperture the SCE I is integrated into the calculation of retinal diffraction
images, in the manner of an apodization effect. And, in the absence of aberrations, the pupil apodization is
chosen twice as narrow as suggested by the SCE I to compute modulation in the retinal light distributions. But
using coherent illumination, a periodic target of rectangular profile and incorporating SCE I as pupil
apodization we got a result where neither the intensity nor the modulation changes indicating absence of SCE I
for a full pupil admitting a single coherent beam.
Keywords: Stiles-Crawford effect, Coherent illumination, Apodization, Rectangular grating.
I.
Introduction
The obliquely incident light on the retina elicits a poor visibility response due to Stiles-Crawford effect
of the first kind (SCE I) [1]. The change in the PSF produced by a gradual shading of the aperture towards its
edge had been described as an apodization [2], because it reduces the height of the surrounding rings in the Airy
disk. However, for the same oblique angle of incidence on the retina the SCE does not discriminate between a
coherent and an incoherent beam [3]. But when an opportunity of interference is offered to the retina by way of
adding another coherent beam to the existing one, retina’s response is no more solely guided by the SCE I [4].
And so long a wavefront slope or an underlying phase is present in the light the retina can ill afford to ignore
that. Hence a visual response in the manner of Stiles-Crawford (SCE I) diminution of effective brightness can be
elicited by generating a wavefront slope at the retina [5]. When the wavefront slope at the retina is zero, the
Poynting vector has also zero obliquity with no reduction in effective brightness. But with increasing wavefront
slope the obliquity of the Poynting vector increases [6]. This is to say that the effective brightness is now
controlled by the Poynting vectors and the SCE I brightness diminution is conditional on the obliquity of the
Poynting vector at the retinal site where photon transduction takes place. The obliquity of the Poynting vectors
somehow kills the interference effect in the retinal plane owing to rapid and irregular phase fluctuations in the
incoherent illumination [6]. With the disappearance of the interference effect the SCE I brightness reduction for
oblique incidence reappears [7]. This also excludes hitherto widely used and implemented effective shading of
the wavefront amplitude before interference [8-13] as a possible explanation for SCE I. By treating the SCE I as
a pupil apodization in the diffraction calculations of the retinal light distributions for a periodic rectangular
target in coherent illumination I got a null result which backs the experimental findings, that unlike incoherent
illumination, the SCE I as an apodization ceases to be effective in case of coherent illumination.
II.
Approach
The change in the point spread function (PSF) produced by a gradual shading of the aperture toward its
edge had been described as an apodization [2], because it reduces the height of the surrounding rings in the Airy
disk. The reasonable assumption of light entering into a cone through a single transverse aperture allows us to
incorporate the SCE I into the computation of retinal light distributions in the manner of an apodizing effect
[14]. In the absence of aberrations this gives the identical prediction for effective retinal diffraction images with
the photoreceptor waveguiding provided one is careful in multiplying the amplitude of the wavefront at each
pupil location with the square root of the SCE I at that pupil location (square root because the SCE I has been
measured for intensity and not amplitude) prior to the transformation leading to the retinal image light
distribution [15]. Moreover unlike incoherent illumination, the non-linearity in intensity for coherent imaging
does not allow the formulation of any general statements about the retinal image [16]. Thus the particular choice
of test target and pupil opening (that too, in the absence of aberrations and defocus) will have its own kind of
imaging characteristics in coherent illumination. What is then the series of transformations in case of coherent
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The Dependence of Spatial Distribution of Excitement of Receptors on Their Stiles-Crawford Effect
imaging ultimately leading to the fundamental imaging property of a human eye which is contained either in the
modulation transfer function in the Fourier domain or in the image intensity distribution of a point object in the
space domain? The amplitude of light in the pupil plane is the Fourier transform of the amplitude in the plane of
the rectangular grating. This distribution represents the spatial frequency content of the amplitude displayed at
distances from the geometrical image of the source proportional to the spatial frequencies.
Ultimately the amplitude distribution of light in the plane of the retina is subjected to a non-linear
transformation, that is, squaring, and the resulting intensity distribution constitutes the retinal image [16].
This I have done by evaluating the modulation, an effective parameter for image quality assessment by
choosing a rectangular wave grating as a test target. When the target, here the periodic rectangular wave grating
is illuminated by spatially coherent light, the amplitude of light coming out of the target plane is displayed as a
diffraction image in the plane of the pupil. It is then filtered, here the SCE I acting as a neutral density wedgeshaped filter. After this it is resynthesized as a distribution of amplitude of light in the retinal plane. Finally the
amplitude is squared at each point in the image giving rise to image intensity distribution leading to modulation.
The approach is to incorporate SCE I in retinal diffraction image computation as an apodization in
coherent illumination and to get results which confirm apodization’s ineffectiveness. First, the Fourier transform
of the amplitude transmittance of a vertical rectangular wave grating with average amplitude , modulation ,
period and duty cycle is taken [17]
(
)
( ) (
)
∑
(
)
to obtain the object spatial frequency spectrum (
)
[(
as ∬
)
(
∑
)
(
∑ , (
(
) ( )
by utilizing the properties of Dirac delta function.
)]
)
(
)
(
)(
)
Multiplying this object Fourier spectrum with the amplitude transfer function
(
)
, (where
) , - which is mathematically identical to the pupil function within a
coordinate scaling operation for spatially coherent illumination, I got the image spatial frequency spectrum as
(
(
)
)
[(
) (
)
∑ * (
)
(
)+]
Inverse transforming this spectrum and again utilizing the properties of Dirac delta function it is found
(
)
( ) (
)
∑
(
)
that the output field is:
(
As the amplitude function
write the upper limit of is such that
by the squared modulus of this output field.
(
)
[(
)
)
by definition, is zero outside the pupil and 1 inside, we can
. Finally, the intensity distribution in the image plane is given
(
∑
)
(
)]
) is found out by evaluating the maximum and minimum intensity at
The modulation (
=0 and
=π respectively.
I have compared the intensity without and with SCE I as pupil apodization for different duty cycle of
the grating. The modulation
is also compared in the absence and presence of SCE I, again for various duty
cycle.
III.
Results And Discussion
It is to be noted that throughout this paper the spatial frequency is normalized with respect to the cut-off spatial
frequency. Thus, for example, when
is mentioned, it means the spatial frequency discussed is just 20 %
of the cut-off spatial frequency for coherent illumination. Similar is the case with reduced distance ( ) In
order to understand the response of modulation to spatial coherence of the entering beams in presence of the
Stiles Crawford effect of the first kind, two categories of relations are studied, first, the variation of intensity
with reduced distance, and next, the response of modulation to different normalized spatial frequencies. This is
done both by modeling Stiles Crawford effect of the first kind as a pupil apodization and without it. To give
richness to the study, different duty cycles are taken into consideration. Again, two types of normalized spatial
frequencies, low and high are chosen.
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The Dependence of Spatial Distribution of Excitement of Receptors on Their Stiles-Crawford Effect
Fig.1: Comparison of intensity for small ω taking α=0.1 & 0.3.
Fig.3: Comparison of intensity for large ω taking α=0.1 & 0.4.
Fig.2: Comparison of intensity for small ω taking α=0.4
Fig.4: Comparison of intensity for large ω taking α=0.7 & 0.8.
Fig.5: Comparison of modulation for α=0.1 & 0.2.
Fig.6: Comparison of modulation for α=0.3 & 0.4.
Fig.7: Comparison of modulation for α=0.6 & 0.7.
Fig.8: Comparison of modulation for α=0.8 & 0.9.
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The Dependence of Spatial Distribution of Excitement of Receptors on Their Stiles-Crawford Effect
The intensity distribution in the images of periodic rectangular wave targets, formed by a human eye
apodized with SCE-I under coherent illumination was computed (from figs 1 – 4), both for a human eye
apodized with the SCE I and in the absence of it for different values of the duty cycle of the periodic target
chosen. It is seen from the figures that for coherent illumination, apodization treatment does not modify the
intensity in each case of the duty cycle for both low and high normalized spatial frequency.
Similarly, the intensity expression is used to find out the modulations in the images plotted graphically
from figs. 5-8. From the graphs it is evident that the modulation changes only by 4 % throughout in almost all
cases of the duty cycles taken here indicating almost absence of directionality for a SCE-I apodized human eye
under coherent illumination over the entire spatial frequency range for a rectangular wave target.
IV.
Conclusion
The result thus arrived at taking into account a rectangular wave periodic grating strengthens the
experimental finding [4-5] of the absence of the traditional SCE-I directionality for the entire human pupil under
coherent illumination. And it raises the fundamental question of how light beams capable of interference, finally
affect retinal image formation differing distinctly from beams not able to interfere.
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